Question: Solve for $x$ : $ 5|x - 5| - 4 = -5|x - 5| + 4 $
Explanation: Add $ {5|x - 5|} $ to both sides: $ \begin{eqnarray} 5|x - 5| - 4 &=& -5|x - 5| + 4 \\ \\ { + 5|x - 5|} && { + 5|x - 5|} \\ \\ 10|x - 5| - 4 &=& 4 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 10|x - 5| - 4 &=& 4 \\ \\ { + 4} &=& { + 4} \\ \\ 10|x - 5| &=& 8 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x - 5|} {{10}} = \dfrac{8} {{10}} $ Simplify: $ |x - 5| = \dfrac{4}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 5 = -\dfrac{4}{5} $ or $ x - 5 = \dfrac{4}{5} $ Solve for the solution where $x - 5$ is negative: $ x - 5 = -\dfrac{4}{5} $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& -\dfrac{4}{5} \\ \\ {+ 5} && {+ 5} \\ \\ x &=& -\dfrac{4}{5} + 5 \end{eqnarray} $ Change the ${ + 5}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{4}{5} {+ \dfrac{25}{5}} $ $ x = \dfrac{21}{5} $ Then calculate the solution where $x - 5$ is positive: $ x - 5 = \dfrac{4}{5} $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& \dfrac{4}{5} \\ \\ {+ 5} && {+ 5} \\ \\ x &=& \dfrac{4}{5} + 5 \end{eqnarray} $ Change the ${ + 5}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{4}{5} {+ \dfrac{25}{5}} $ $ x = \dfrac{29}{5} $ Thus, the correct answer is $x = \dfrac{21}{5} $ or $x = \dfrac{29}{5} $.